$Q$-divisible modules

Efraim P. Armendariz
1971 Canadian mathematical bulletin  
1. Introduction. Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6] . In a recent paper [12] , Wei called a (left) .R-module M divisible in case Hom B (Q, 7V)#0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an i?-module a Q-divisible i?-module. As shown in [12] , the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5] . It
more » ... s that every i?-module M contains a (unique) maximal ô-divisible submodule D(M) such that M/D(M) contains no nonzero g-divisible submodule. Moreover, the class D contains all injective i?-modules and hence contains the torsion class D 0 generated by the injective i?-modules. In general D and D 0 are distinct, but in some instances coincidence of these classes occurs. In this note we examine some of these situations as well as some relationship between the class D and the class of jR-modules with zero singular submodule. (As in [9], we call modules with zero singular submodule nonsingular and if the (left) singular ideal of R is zero then R is a nonsingular ring.) In §2 we characterize rings for which every g-divisible module is injective, nonsingular rings for which every nonsingular g-divisible module is injective, and finite-dimensional nonsingular rings for which every g-divisible i?-module is a factor of an injective i?-module. In §3, some examples are given related to the classes D and D 0 .
doi:10.4153/cmb-1971-087-9 fatcat:l5mgvrfyhrcx3g65mfjlems6fy